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The given function modeling the rabbit population is:

\[
P(t) = \frac{200t}{1 + 5t}
\]
where \( t \) is the time in months.

We are tasked with understanding how the population \( P(t) \) behaves as \( t \) (time) increases.

### Step 1: Analyzing Long-term Behavior
As \( t \to \infty \), the term \( 5t \) dominates over the constant 1 in the denominator. Therefore, the function behaves like:

\[
P(t) \approx \frac{200t}{5t} = 40
\]

Thus, as time \( t \) increases indefinitely, the population approaches a limiting value of 40.

### Step 2: Interpretation
The function represents a rational function with a horizontal asymptote at \( P(t) = 40 \). This means that the rabbit population increases initially but eventually stabilizes and levels off at a population of 40 rabbits as time progresses.

### Conclusion:
As time increases, the population of rabbits approaches a stable value of 40. This describes logistic-type growth where the population grows quickly at first but slows down and reaches a plateau.

Would you like further details or clarifications on this?

### Related Questions:
1. How would you find the time at which the population reaches half of its limiting value?
2. What is the population after 5 months?
3. How does the function behave for small values of \( t \)?
4. Can you find the time at which the population reaches 30 rabbits?
5. How would the model change if the initial growth rate were slower?

### Tip:
Understanding the long-term behavior of rational functions often involves finding horizontal asymptotes, which describe the steady-state value of a population or system in applied contexts.

The function P(t) models the population of rabbits on a farm and is given by P(t) = (200t) / (1 + 5t) for t ≥ 0, where t is measured in months since the start of the year. Which of the following describes the population of the rabbits as time increases?

Analyze the behavior of the rabbit population function P(t) = (200t) / (1 + 5t) as time increases. Learn how to determine the population's limiting value as t approaches infinity using the concept of horizontal asymptotes. Suitable for high school students in Grades 10-12.

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